How many 2 x 2 matrices are there with entries from the set {0, 1, ..., k} in which there are no zero rows and no zero columns.

My solution

No. of matrices in which there are zero rows (Assume that the first row is a zero row. There are ways of choosing the elements of the second row. Since the second row could as well be the zero row, we get ways. Since we have counted the zero matrix twice, we must subtract 1.)

No. of matrices in which there are zero columns

No. of matrices in which there are both zero rows and zero columns ( matrices each with one non-zero element and 1 zero matrix.)

No. of matrices in which there are either zero rows or zero columns

Total no. of matrices

No. of matrices in which there are no zero rows and no zero columns

Could someone please check if my solution is correct? Thanks.